STEAM CAD and Structural Geometry

STEAM CAD and Structural Geometry: A Generative Framework for N-Dimensional Polyhedral Invariants

Abstract:

• Briefly introduce:
  
  

  • STEAM CAD as a hierarchical generative system
    
    

  • Structural Geometry and its four disciplines
    
    

  • The generalized Euler-like polyhedral invariants
    
    

• Emphasize originality, generative power, and cross-dimensional application
  
  

1. Introduction

• Context: limitations of classical geometry and topology in generative applications
  
  

• Motivation: need for a system linking data structures, generative design, and physical realization
  
  

• Objectives: formalize STEAM CAD, define Structural Geometry, generalize Euler’s formula
  
  

2. Background / Related Work

• Euler’s polyhedron formula and its classical applications
  
  

• Overview of analytic geometry (Descartes) and algebraic geometry (Fermat)
  
  

• Computational geometry and generative design systems (brief survey)
  
  

3. STEAM CAD Architecture

• Four-level hierarchical model: Departments → Disciplines → Categories → Objects
  
  

• Tables/matrices as generative combinatorial engines
  
  

• Algorithmic rules for producing structures
  
  

• Visual schematic of hierarchy (optional diagram)
  
  

4. Structural Geometry and Its Four Disciplines

• 2×2 table combining geometry and topology → four structural disciplines
  
  

• Definition of each discipline: Structural Metageometry (SMG), Structural Geometric Topology (SGT), Structural Topological Geometry (STG), Structural Metatopology (SMT)
  
  

• Recursive generation of categories and objects (4×4 tables, etc.)
  
  

• Examples of generated structures, potentially 3D printable
  
  

5. Euler’s Revised Polyhedron Formula and N-Dimensional Invariants

• Classical Euler formula review (V - E + F = 2)
  
  

• Carpenter’s generalization: V - E + F - S = 1
  
  

• Hierarchical formulae: V=1, V-E=1, V-E+F=1
  
  

• Extension to N-D polytopes, invariants for zero genus
  
  

• Embedding in Pascal’s Triangle for simplexes
  
  

• Implications for Structural Geometry and STEAM CAD outputs
  
  

6. Applications and Implications

• Generative design and 3D printing
  
  

• Computational mathematics and topology
  
  

• Mathematical art
  
  

• Cross-disciplinary implications
  
  

7. Discussion

• Comparison to historical mathematical contributions (Descartes, Fermat, Euler)
  
  

• Novelty, rigor, and potential impact
  
  

• Limitations and areas for further formalization
  
  

8. Conclusion

• STEAM CAD + Structural Geometry + Euler generalization forms a complete, generative, mathematically rigorous system
  
  

• Reiterate paradigm-shifting nature and cross-disciplinary relevance

9. The STEAManufacturing Ecosystem

• STEAM CAD as part of a three-software pipeline: STEAM CAD → Great Stella → Bambu Studio → 3D Printer
  
  

• STEAM CAD produces data structures/concepts
  
  

• Great Stella visualizes concepts; STL export for fabrication
  
  

• Bambu Studio generates printer code for physical realization
  
  

• Current limitation: graphic design algorithms not automated, rely on user expertise
  
  

• Significance: complete conceptual-to-physical pipeline

.

References / Appendix

• Include tables, diagrams, and pseudocode for STEAM CAD operations
  
  

• Optional: 3D visualizations or links to interactive demonstrations

Abstract

We introduce STEAM CAD, a hierarchical generative system for mathematics, design, and physical realization, and its foundational framework, Structural Geometry. STEAM CAD models an idealized academic structure across four levels—departments, disciplines, categories, and objects—where components at each level recursively generate higher-level structures. Structural Geometry is organized into four interrelated disciplines—Structural Metageometry, Structural Geometric Topology, Structural Topological Geometry, and Structural Metatopology—arising from combinatorial interactions between geometric and topological inputs. This framework produces data structures that are algorithmically generative, enabling the creation of both abstract and physically realizable forms, including 3D-printed constructions.

We further generalize Euler’s polyhedron formula to higher-dimensional polytopes:

V−E+F−S=1

where V is the number of vertices, E edges, F faces, and S solids, yielding a universal invariant of 1 for any geometric structure of positive genus. These formulas extend recursively through n-dimensional polytopes and can be embedded in Pascal’s Triangle for simplexes, providing a consistent invariant across dimensions.

Together, STEAM CAD, Structural Geometry, and the generalized Euler invariants constitute a generative, mathematically rigorous, and cross-disciplinary system, offering a novel paradigm for the creation, classification, and realization of geometric and topological structures.

Introduction

The interplay between geometry, topology, and algebra has historically driven major advances in mathematics. Cartesian analytic geometry bridged algebra and Euclidean geometry, while Fermat’s contributions to algebraic geometry connected equations with spatial forms. Euler’s polyhedron formula established fundamental topological invariants, providing deep insights into the combinatorial properties of convex polyhedra. Yet, these classical systems operate primarily in an abstract symbolic domain, producing descriptive rather than generative structures.

STEAM CAD introduces a novel paradigm: a hierarchical, generative system that converts abstract data structures into physically realizable geometric and topological structures. Modeled on an idealized academic architecture, STEAM CAD organizes knowledge into four levels: departments, disciplines, categories, and objects. Each level is defined by the recursive combination of components from the level below, allowing a finite set of primitives to generate an infinite variety of structures.

Within this framework, Structural Geometry emerges as a new branch of mathematics, composed of four disciplines arising from combinatorial interactions between geometric and topological inputs. These disciplines—Structural Metageometry, Structural Geometric Topology, Structural Topological Geometry, and Structural Metatopology—form the foundational rules for generative construction, providing both algorithmic rigor and cross-domain applicability.

We extend the theoretical foundation of Structural Geometry through a generalization of Euler’s polyhedron formula, producing an invariant applicable to n-dimensional polytopes:

V−E+F−S=1

where V is the number of vertices, E is the number of edges, F is the number of faces, and S is the number of solids. In this way the definition of a polyhedron as a shell used by Euler and the definition of a polyhedron as a solid used by Euclid are combined to produce a novel Polyhedron Formula. 

This invariant, along with the hierarchical formulas V=1, V−E=1, and V−E+F=1, establishes a recursively consistent framework for polyhedral structures of any dimension and zero genus. Embedded in Pascal’s Triangle for simplexes, these invariants unify combinatorial, geometric, and topological principles, providing a mathematically rigorous foundation for the generative capabilities of STEAM CAD.

In this paper, we present the architecture, disciplines, and mathematical underpinnings of STEAM CAD, demonstrate its capacity for generative construction, and discuss the implications of generalized Euler invariants for mathematics, computational design, and physical realization. Together, these contributions constitute a novel paradigm at the intersection of mathematics, computation, and design, offering both theoretical rigor and practical generative power.

3. STEAM CAD Hierarchy and Generative Architecture

3.1 Overview

STEAM CAD is a hierarchical, generative system designed to produce both abstract and physically realizable structures from a finite set of primitives. Its architecture is modeled on an idealized conception of academia, comprising four hierarchical levels:

1. Departments – The highest-level organizational units, representing broad domains of STEAM knowledge. (e.g. Mathematics)
   
   

2. Disciplines – Components of departments and defined by the categories of objects they study at the level below them. (e.g  geometry)
   
   

3. Categories of Objects – Collections of related objects by class that function to define disciplines. (e.g. polyhedra)
   
   

4. Objects – The atomic units at the lowest level, representing fundamental geometric or topological primitives. (e.g. cubes)
   
   

At each level, the objects of the lower level function as components for the construction of the next-higher level. This recursive arrangement enables STEAM CAD to generate complex structures from simple primitives, ensuring both scalability and reproducibility.

3.2 Generative Tables and Component-Structure Mapping

STEAM CAD employs partition matrices or tables that function as analog computers with inputs, operations, and outputs to specify how components combine into structures. Thus it is fundamentally a computational system defined by structuralism. Each table defines:

• Input Components: e.g. elements like points, lines, polygons, polyhedra for geometry are combined with
  
  

• Input Structures: like knots, links, braids, weaves for topology

• Output Structures: New higher-level compound categories, or objects are produced by applying the generative rules.
  
  

Example: At the discipline level, a 2×2 table combines geometry and topology as inputs to produce the four structural disciplines: Table 1

• SMG (Structural Metageometry): Geometric structures from geometric components
  
  

• SGT (Structural Geometric Topology): Topological structures from geometric components
  
  

• STG (Structural Topological Geometry): Geometric structures from topological components
  
  

• SMT (Structural Metatopology): Topological structures from topological components
  
  

This generative table framework is applied recursively at the category and object levels. For instance, in Structural Geometric Topology (SGT), a 4×4 table may combine:

• Component inputs: points, lines, polygons, polyhedra
  
  

• Structure inputs: knots, links, braids, weaves
  
  

The resulting 16 compound categories define the discipline at a finer granularity. At the object level, additional tables combine objects as components and structures, ultimately enabling graphical design for 3D printing or hand construction (Table 2).

3.3 Algorithmic Rules and Recursive Generativity

STEAM CAD formalizes algorithmic rules that govern the combination of components at each hierarchical level. Key features include:

• Recursive Generativity: Each level draws from the output of the previous, enabling infinite structural variation from finite primitives.
  
  

• Component-Structure Duality: Components are both building blocks and constraints, ensuring that outputs maintain mathematical and structural consistency.
  
  

• Scalability: The system supports arbitrary depth, from simple objects to high-level composite structures.
  
  

• Physical Realization: Algorithmically generated structures can be directly translated into 3D-printed models or manually constructed forms, bridging the abstract and the physical.
  
  

3.4 Significance

This hierarchical, generative architecture positions STEAM CAD as more than a design tool: it is a foundational system for Structural Geometry, formalizing:

• The recursive nature of complex structures
  
  

• The mapping between geometric and topological inputs and outputs
  
  

• A framework for N-dimensional invariants (to be discussed in Section 5)
  
  

By combining structured hierarchies, algorithmic rules, and recursive tables, STEAM CAD provides the engine for generating novel structures, while Structural Geometry supplies the formal mathematical framework ensuring rigor and consistency.

4. Structural Geometry and the Four Structural Disciplines

4.1 Overview

Structural Geometry (SG) is a foundational component of STEAM CAD, formalizing the rules by which geometric and topological components generate higher-order structures. Unlike classical Euclidean geometry, which emphasizes axioms, proofs, and theorems, Structural Geometry is algorithmic and generative, designed to produce both abstract data structures and physically realizable forms.

SG is organized into four interrelated disciplines, each arising from the combinatorial interaction between geometric and topological inputs:

1. Structural Metageometry (SMG): Geometric structures constructed from geometric components.
   
   

2. Structural Geometric Topology (SGT): Topological structures constructed from geometric components.
   
   

3. Structural Topological Geometry (STG): Geometric structures constructed from topological components.
   
   

4. Structural Metatopology (SMT): Topological structures constructed from topological components.
   
   

These disciplines are generated recursively through hierarchical tables, from the discipline level down to categories and objects.

4.2 Recursive Table Framework

Each SG discipline is defined using component-structure tables that specify how lower-level elements combine to form higher-level constructs.

4.2.1 Discipline Level (See Table 1)

Interpretation:

• The rows represent the type of input structures (Geometric or Topological).
  
  

• The columns represent component inputs also geometric and topological
  
  

• The output cells in the output array denote the resulting structural disciplines.
  
  

This 2×2 table forms the conceptual backbone of Structural Geometry, capturing the four fundamental generative interactions between geometry and topology.

4.2.2 Category Level (See Table 2)

Within a discipline such as SGT, categories further specify the structural rules:

• Component inputs (rows): points, lines, polygons, polyhedra
  
  

• Structure inputs (columns): knots, links, braids, weaves
  
  

• 
  Each cell represents a compound category—a specific combination of geometric and topological elements.
  
  

• For SGT, this yields 16 distinct categories, which define the discipline’s combinatorial landscape.
  
  

4.2.3 Object Level (See Table 3)

• At the object level, each compound category is further refined into atomic units, the objects of structural geometric topology. For example, specific polygons like triangles, squares, and pentagons or cubes, tetrahedra, and dodecahedra for polyhedra (Table 3).
  
  

• Additional tables specify how these objects combine, producing higher-order outputs suitable for graphical design, computational modeling, or 3D printing.
  
  

• This recursive process ensures that every generated structure is mathematically consistent and physically realizable.
  
  

4.3 Algorithmic Generativity

The generative process in Structural Geometry is governed by algorithmic rules:

1. Components serve as both building blocks and constraints, ensuring coherence.
   
   

2. Structures guide combination, producing consistent outputs across hierarchical levels.
   
   

3. Recursion allows infinite generative depth, enabling complex structures from a finite set of primitives.
   
   

4. Output is both abstract and physical, bridging computational mathematics and tangible realization.
   
   

4.4 Significance and Applications

Structural Geometry provides:

• A formal mathematical foundation for STEAM CAD’s generative outputs.
  
  

• Scalable and recursive combinatorial rules that operate at multiple hierarchical levels.
  
  

• The ability to integrate geometry and topology in ways that extend beyond classical mathematics.
  
  

• Practical applicability in:
  
  

  • 3D printing of complex polyhedral or topological structures
    
    

  • Computational design and modeling
    
    

  • Mathematical art and visualization
    
    

  • Cross-disciplinary educational tools
    
    

Summary: By defining four recursive, algorithmically generative disciplines, Structural Geometry establishes a new mathematical paradigm—one that is simultaneously foundational, generative, and physically realizable.

5. Euler’s Revised Polyhedron Formula and N-Dimensional Invariants

5.1 Classical Background

Euler’s polyhedron formula is a foundational result in topology and combinatorial geometry:

V−E+F=2

where V is the number of vertices, E is the number of edges, and F is the number of faces of a convex polyhedron. This invariant provides a critical link between geometry and topology, offering a combinatorial property that holds for all convex polyhedra.

While powerful, Euler’s formula is traditionally limited to three-dimensional convex polyhedra. Extensions to higher-dimensional polytopes exist, but a universally recursive formulation connecting geometry, topology, and generative structures has remained elusive.

5.2 Carpenter’s Generalization

Carpenter introduces a revised polyhedron formula that integrates Structural Geometry with Euclidean geometry and generalizes Euler’s invariant to higher dimensions:

V−E+F−S=1

where:

• V = number of vertices
  
  

• E = number of edges
  
  

• F = number of faces
  
  

• S = number of Euclidean solids

These become the components of  geometric and topological structures such as points, lines, polygons, and polyhedra in structural geometry.

This formula produces a universal invariant of 1 for any geometric structure of zero genus, providing a recursively consistent metric that can be applied across arbitrary N-dimensional polytopes.

5.3 Hierarchical N-Dimensional Invariants

The revised formula is part of a hierarchy of invariants:

V=1, V-E=1, V - E + F = 1 (Table 4)

(Table 4)

• Each formula corresponds to a specific dimensional level: points (0D), lines (1D), polygons (2D), polyhedra (3D), and higher-dimensional analogues.
  
  

• These invariants extend recursively to N dimensions, defining a consistent topological and combinatorial property for polytopes at every level.
  
  

5.3.1 Embedding in Pascal’s Triangle

Carpenter demonstrates that these invariants can be placed within Pascal’s Triangle for simplexes, where each entry corresponds to a component count in the generative hierarchy. The universal invariant of 1 is preserved across dimensions, unifying combinatorial, geometric, and topological principles within a single framework (Table 5)

(Table 5)

5.4 Integration with Structural Geometry

• The generalized Euler invariants serve as mathematical constraints within the generative tables of Structural Geometry.
  
  

• At the object level generated structures adhere to these invariants, ensuring both mathematical consistency and physical realizability.
  
  

• This integration transforms STEAM CAD from a purely algorithmic system into a mathematically rigorous generative engine, capable of producing structures that are both novel and invariantly well-formed.
  
  

5.5 Significance

1. Foundational: Provides a mathematically rigorous backbone for Structural Geometry.
   
   

2. Generative: Guides the recursive creation of structures across multiple dimensions.
   
   

3. Universal: The invariant applies to any geometric structure of zero genus, including higher-dimensional polytopes.
   
   

4. Cross-disciplinary: Bridges combinatorics, geometry, topology, and generative design.
   
   

5. Practical: Supports the generation of 3D-printable structures and complex models, ensuring all zero genus outputs satisfy the invariant.
   
   

Summary: Carpenter’s revised Euler formula and N-dimensional invariants extend the classical foundations of topology into a generative, algorithmically consistent, and physically realizable system, providing Structural Geometry with a formal, unifying framework.

The significance of these results to structural cosmology are explored at: Albert P. Carpenter - E Pluribus Unum 

6. Applications and Implications

6.1 Applications

The integration of STEAM CAD, Structural Geometry, and the generalized Euler invariants enables a broad spectrum of applications across mathematics, design, and physical construction:

1. Generative Design and 3D Printing
   
   

   • STEAM CAD produces complex geometric and topological structures that can be directly fabricated using 3D printing.
     
     

   • The recursive tables and hierarchical generative rules ensure all printed structures adhere to mathematical invariants, preserving structural integrity and geometric consistency.

   • The standardized architecture of STEAM CAD ensures interoperability accross STEAM departments, disciplines, categories, and objects.
     
     

2. Computational Mathematics and Topology
   
   

   • Structural Geometry formalizes algorithmic generation of structures while maintaining mathematical rigor.
     
     

   • N-dimensional invariants provide a universal metric for evaluating and classifying geometric and topological forms, offering new avenues for research in higher-dimensional combinatorics, topology, and polytope theory.
     
     

3. Mathematical Art and Visualization
   
   

   • Generated structures can serve as mathematical artworks, combining aesthetic design with formal invariants.
     
     

   • Interactive visualizations of recursive structures, tables, and 3D models make complex mathematics accessible and tangible.
     
     

4. Educational and Cross-Disciplinary Tools
   
   

   • STEAM CAD provides a hands-on framework for teaching geometry, topology, and combinatorial mathematics.
     
     

   • By bridging algorithmic generation and physical realization, the system fosters STEAM-based learning, connecting abstract theory with creative experimentation.

5. Implementation as Software

• STEAM CAD has the potential to be implemented as a software program as described here.
  
  

6.2 Implications for Mathematics

1. Novel Mathematical Paradigm
   
   

   • Structural Geometry represents a shift from purely descriptive mathematics to generative mathematics, producing new structures algorithmically while maintaining formal invariants.
     
     

2. Extension of Classical Geometry and Topology
   
   

   • By generalizing Euler’s formula to N dimensions and integrating it with recursive generative rules, Carpenter provides a new combinatorial and topological framework applicable to higher-dimensional polytopes and complex structures.
     
     

3. Bridging Theory and Practice
   
   

   • The system demonstrates that abstract mathematical concepts can directly inform physical construction, establishing a feedback loop between theory, computation, and fabrication.
     
     

• Algorithmic Generativity as a Mathematical Tool
  
  

  • STEAM CAD exemplifies how algorithmic processes can become a form of mathematical reasoning, producing structures that embody both combinatorial and topological invariants.
    
    

• Cross-Disciplinary Significance
  
  

  • Beyond mathematics, the system impacts design, architecture, engineering, and education, illustrating the potential of generative frameworks to unify theory and practice.
    
    

• Potential Future Work
  
  

  • Formal proofs of invariants for higher-dimensional polytopes
    
    

  • Integration with computational geometry software for automated verification
    
    

  • Expansion of Structural Geometry to incorporate additional mathematical domains, such as differential geometry or category theory
    
    

  • Interactive educational platforms to teach Structural Geometry and STEAM CAD principles
    
    

6.4 Summary

STEAM CAD, Structural Geometry, and the generalized Euler invariants together form a cohesive, generative, and mathematically rigorous system. The framework:

• Provides hierarchical, recursive rules for generating structures
  
  

• Maintains universal invariants across dimensions
  
  

• Bridges abstract theory and physical realization
  
  

• Opens pathways for cross-disciplinary applications in mathematics, design, and education
  
  

This combination represents a paradigm shift in generative mathematics, offering both theoretical novelty and practical utility.

7. Conclusion

This paper introduces STEAM CAD, a hierarchical generative system that formalizes the creation of Structural Geometry, and presents a generalization of Euler’s polyhedron formula to N-dimensional invariants. Together, these contributions establish a novel mathematical framework that bridges abstract combinatorial theory, topology, and physical realization.

The hierarchical architecture of STEAM CAD—departments, disciplines, categories, and objects—enables recursive generativity, allowing finite primitives to produce complex structures at multiple scales. Structural Geometry organizes these generative processes into four interrelated disciplines, formalizing the combinatorial interactions between geometric and topological components.

Carpenter’s generalized Euler formula,

V−E+F−S=1

and the associated hierarchical invariants extend classical topological results to arbitrary dimensions, providing a universal metric for structure formation. When integrated with the recursive tables for the objects of Structural Geometry, these invariants ensure mathematical consistency across all generated outputs, from abstract data structures to physically realized 3D models.

The framework presented here demonstrates that generative, algorithmic systems can constitute a mathematically rigorous methodology, expanding the boundaries of both mathematics and applied design. Beyond theoretical significance, STEAM CAD and Structural Geometry offer practical applications in 3D printing, computational design, mathematical art, and STEAM education, illustrating a paradigm shift from descriptive to generative mathematics.

In summary, this work provides:

1. A formal, hierarchical architecture for generative mathematics.
   
   

2. Four interrelated disciplines of Structural Geometry enabling recursive and scalable structure creation.
   
   

3. A generalized Euler formula and N-dimensional invariants applicable across all polyhedral structures.
   
   

4. A framework for bridging abstract mathematical theory with physical realization.
   
   

Together, these contributions establish a foundational system for generative mathematics, offering both theoretical novelty and practical utility, and opening avenues for further research, computational exploration, and interdisciplinary application.

8. The STEAManufacturing Ecosystem

STEAM CAD if implemented as software would be one of three integrated software programs that together form a complete ecosystem for STEAManufacturing—the transformation of abstract concepts into tangible, physical structures (Figure 1).

Workflow Overview:

1. STEAM CAD – Concept Generation (status: does not yet exist except in analog form)
   
   

   • Produces data structures (concepts) via Structural Geometry and algorithmic generativity.
     
     

   • Defines hierarchical relationships between departments, disciplines, categories, and objects.
     
     

   • Serves as the idea engine, providing the conceptual foundation for subsequent steps.
     
     

2. Great Stella (Robert Webb) – Graphic Design (status: exists)
   
   

   • Translates concepts from STEAM CAD into visual and geometric representations.
     
     

   • Exports STL files for 3D modeling.
     
     

   • At this stage, the graphic design relies on user expertise, as algorithms for automated visualization do not yet exist.
     
     

3. Bambu Studio – Slicer and Code Generation (status: exists)
   
   

   • Converts STL files into printer-ready code for 3D fabrication.
     
     

   • Realizes the physical manifestation of the original STEAM CAD concept.
     
     

Significance:

• The ecosystem creates a continuous chain from idea to physical object, bridging abstract mathematics, computational design, and fabrication.
  
  

• STEAM CAD initiates the process, but full automation of graphic design remains a frontier for development as yet there are no algorithms that specify design procedures.
  
  

• Thus, users remain central to interpreting and visualizing data structures, highlighting the creative and human-driven aspect of STEAManufacturing.
  
  

(Figure 1)

References

1. Euler, L. Elementa Doctrinae Solidorum. Novi Commentarii Academiae Scientiarum Petropolitanae, 1758.
   
   

2. Descartes, R. La Géométrie. 1637.
   
   

3. Fermat, P. Ad Locos Planos et Solidos. 1637.
   
   

4. Coxeter, H.S.M. Regular Polytopes. 3rd Edition, Dover Publications, 1973.
   
   

5. Grünbaum, B. Convex Polytopes. 2nd Edition, Springer, 2003.
   
   

6. Hatcher, A. Algebraic Topology. Cambridge University Press, 2002.
   
   

7. Carpenter, A.P. STEAM CAD and Structural Geometry: Hierarchical Generative Systems for N-Dimensional Polyhedral Invariants. Manuscript in preparation, 2025.
   
   

8. Carpenter, A.P. Structural Geometry and Algorithmic Generativity. Website: www.structuralgeometry.com, 2025.
   
   

9. Weisstein, E.W. Euler Characteristic. MathWorld – A Wolfram Web Resource.
   
   

10. Felsner, S., & Ziegler, G.M. Geometric Combinatorics. Springer, 2011.
    
    

Note: Any unpublished material (e.g., Carpenter’s manuscripts or STEAM CAD demonstrations) can be hosted online with a DOI or institutional repository for verification. Written with ChatGPT.